A266695 -id:A266695 - cp's OEIS frontend

A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 46, 46, 16, 1, 1, 32, 146, 230, 146, 32, 1, 1, 64, 454, 1066, 1066, 454, 64, 1, 1, 128, 1394, 4718, 6902, 4718, 1394, 128, 1, 1, 256, 4246, 20266, 41506, 41506, 20266, 4246, 256, 1, 1, 512, 12866, 85310, 237686, 329462, 237686, 85310, 12866, 512, 1

Offset: 0

Ralf Stephan, Oct 27 2004

B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructible from its row and column sums. [Brewbaker]
B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k }. [Launois]
T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan, Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)
T(n,k) is also the number of acyclic orientations of the complete bipartite graph K_{n,k}. - Vincent Pilaud, Sep 15 2020
When indexed as a triangular array, T(n,k) is the number of permutations of [n] in which 1 is in position k and the excedance entries are precisely the entries to the left of 1. See link. - David Callan, Dec 12 2021
T(n,k) is also the number of max-closed relations between an ordered n-element set and an ordered k-element set (see the paper by Jeavons and Cooper 1995). -  Don Knuth, Feb 12 2024

			Square array begins:
  1,  1,   1,    1,     1,      1, ...
  1,  2,   4,    8,    16,     32, ...
  1,  4,  14,   46,   146,    454, ...
  1,  8,  46,  230,  1066,   4718, ...
  1, 16, 146, 1066,  6902,  41506, ...
  1, 32, 454, 4718, 41506, 329462, ...
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, arXiv:2104.13654 [math.CO], 2021. See Table 1 (a) p. 4.
Beáta Bényi, Advances in Bijective Combinatorics, Ph. D. Dissertation, Doctoral School of Mathematics and Computer Science, University of Szeged, Bolyai Institute, 2014. See Table 1.
Beáta Bényi and Peter Hajnal, Combinatorics of poly-Bernoulli numbers, arXiv:1510.05765 [math.CO], 2015.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016.
Beata Benyi and Peter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020.
Beáta Bényi and Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], 2017.
Beáta Bényi and José Luis Ramírez, On q-poly-Bernoulli numbers arising from combinatorial interpretations, arXiv:1909.09949 [math.CO], 2019.
Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers - the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.
Beáta Bényi and José Luis Ramírez, Poly-Cauchy Numbers of the Second Kind-the Combinatorics Behind, Enumerative Comb. Appl. (2022) Vol. 2, No. 1, Art. #S2R1.
Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.
David Callan, Permutations whose excedance positions are those before 1
Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 12.
Peter G. Jeavons and Martin C. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1995), 327-339.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (0.1).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
Stéphane Launois, Combinatorics of H-primes in quantum matrices, Journal of Algebra, Volume 309, Issue 1, 2007, Pages 139-167.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
H. A. Witek, G. Mos and C.-P. Chou, Zhang-Zhang Polynomials of Regular 3-and 4-tier Benzenoid Strips, MATCH Commun. Math. Comput. Chem. 73 (2015) 427-442.
Wikipedia, Acyclic orientation

Rows 0..9 are A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.
Main diagonal is A048163. Another diagonal is A188634.
Antidiagonal sums are in A098830.
Cf. A019538, A210381, A266695.

A:= (n, k)-> add(Stirling2(n+1, i+1)*Stirling2(k+1, i+1)*
             i!^2, i=0..min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Jan 02 2016

T[n_, k_] := Sum[(-1)^(j+n)*(1+j)^k*j!*StirlingS2[n, j], {j, 0, n}]; Table[ T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)

T(n,k)=sum(j=0,n,(j+1)^k*sum(i=0,j,(-1)^(n+j-i)*binomial(j,i)*(j-i)^n))

T(n,k)=sum(j=0,min(n,k), j!^2*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2)); \\ Michel Marcus, Mar 05 2017

pB(k, n) = (-1)^n * Sum[i=0..n, (-1)^i * i! * Stirling2(n, i) / (i+1)^k ].
E.g.f.: e^(x+y) / [e^x + e^y - e^(x+y)].
T(n, k) = Sum_{j=0..n} (j+1)^k*Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n. - Paul D. Hanna, Nov 04 2004
n-th row of the array = row sums of n-th power of triangle A210381. - Gary W. Adamson, Mar 21 2012

A267383 Number A(n,k) of acyclic orientations of the Turán graph T(n,k); square array A(n,k), n>=0, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 6, 14, 1, 1, 1, 2, 6, 18, 46, 1, 1, 1, 2, 6, 24, 78, 230, 1, 1, 1, 2, 6, 24, 96, 426, 1066, 1, 1, 1, 2, 6, 24, 120, 504, 2286, 6902, 1, 1, 1, 2, 6, 24, 120, 600, 3216, 15402, 41506, 1

Offset: 0

Alois P. Heinz, Jan 13 2016

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

Conjecture: In general, column k > 1 is asymptotic to n! / ((k-1) * (1 - log(k/(k-1)))^((k-1)/2) * k^n * (log(k/(k-1)))^(n+1)). - Vaclav Kotesovec, Feb 18 2017

			Square array A(n,k) begins:
  1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    1,    1,    1,    1,    1, ...
  1,    2,    2,    2,    2,    2,    2, ...
  1,    4,    6,    6,    6,    6,    6, ...
  1,   14,   18,   24,   24,   24,   24, ...
  1,   46,   78,   96,  120,  120,  120, ...
  1,  230,  426,  504,  600,  720,  720, ...
  1, 1066, 2286, 3216, 3720, 4320, 5040, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
Wikipedia, Acyclic orientation
Wikipedia, Turán graph

Columns k=1-10 give: A000012, A266695, A266858, A267384, A267385, A267386, A267387, A267388, A267389, A267390.
Main diagonal gives A000142.
A(2n,n) gives A033815.
A(n,ceiling(n/2)) gives A161132.
Bisection of column k=2 gives A048163.
Trisection of column k=3 gives A370961.
a(n^2,n) gives A372084.

A:= proc(n, k) option remember; local b, l, q; q:=-1;
       l:= [floor(n/k)$(k-irem(n,k)), ceil(n/k)$irem(n,k)];
       b:= proc(n, j) option remember; `if`(j=1, (q-n)^l[1]*
             mul(q-i, i=0..n-1), add(b(n+m, j-1)*
             Stirling2(l[j], m), m=0..l[j]))
           end; forget(b);
       abs(b(0, k))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = Module[{ b, l, q}, q = -1; l = Join[Array[Floor[n/k] &, k - Mod[n, k]], Array[ Ceiling[n/k] &, Mod[n, k]]]; b[nn_, j_] := b[nn, j] = If[j == 1, (q - nn)^l[[1]]*Product[q - i, {i, 0, nn - 1}], Sum[b[nn + m, j - 1]*StirlingS2[l[[j]], m], {m, 0, l[[j]]}]]; Abs[b[0, k]]]; Table[Table[A[n, 1 + d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

Original entry on oeis.org

1, 2, 14, 230, 6902, 329462, 22934774, 2193664790, 276054834902, 44222780245622, 8787513806478134, 2121181056663291350, 611373265185174628502, 207391326125004608457782, 81791647413265571604175094, 37109390748309009878392597910, 19192672725746588045912535407702

Offset: 1

N. J. A. Sloane, R. K. Guy

nonn

a(n) is also the number of max-closed relations on an ordered n-element domain (see the paper by Jeavons and Cooper, 1995). - Don Knuth, Feb 12 2024

			1
1 + 1 = 2
1 + 9 + 4 = 14
1 + 49 + 144 + 36 = 230
1 + 225 + 2500 + 3600 + 576 = 6902
... - _Philippe Deléham_, May 30 2015

Lovasz, L. and Vesztergombi, K.; Restricted permutations and Stirling numbers. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 731-738, Colloq. Math. Soc. Janos Bolyai, 18, North-Holland, Amsterdam-New York, 1978.
K. Vesztergombi, Permutations with restriction of middle strength, Stud. Sci. Math. Hungar., 9 (1974), 181-185.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.
Peter G. Jeavons and Martin C. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1995), 327-339.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.
H.-K. Kim et al., Poly-Bernoulli numbers and lonesum matrices, arXiv:1103.4884 [math.CO], 2011.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).

Main diagonal of array A099594.

Cf. A220181, A266695, A306209.

Table[Sum[((k-1)!)^2*StirlingS2[n,k]^2,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Jun 21 2013 *)

a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1)*(m-1)!*x^m/prod(k=1, m-1, 1+m*k*x+x*O(x^n))), n)) \\ Paul D. Hanna, Jan 05 2013
for(n=1,20,print1(a(n),", "))

Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
a(n)=sum(k=1,n,(-1)^(n-k)*k^(n-1)*(k-1)!*Stirling2(n-1, k-1))
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 06 2013

a(n) = sum(k=1, n, (k-1)!^2*stirling(n,k,2)^2); \\ Michel Marcus, Jun 22 2018

E.g.f. (with offset 0): Sum((1-exp(-(m+1)*z))^m, m=0..oo)
O.g.f.: Sum_{n>=1} n^(n-1) * (n-1)! * x^n / Product_{k=1..n-1} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... . - Vaclav Kotesovec, Jun 21 2013
a(n) ~ 2*sqrt(Pi) * n^(2*n-3/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n-1)). - Vaclav Kotesovec, May 13 2014
a(n+1) = Sum_{k = 0..n} A163626(n,k)^2. - Philippe Deléham, May 30 2015
a(n) = A306209(2n-2,n-1). - Alois P. Heinz, Feb 01 2019
a(n) = A266695(2n-2). - Alois P. Heinz, Apr 17 2024

Entry revised by N. J. A. Sloane, Jul 05 2012

A212084 Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

Original entry on oeis.org

1, 1, -1, 0, 1, -4, 6, -3, 0, 1, -9, 36, -75, 78, -31, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0, 1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, 5552680, -6796926, 4787174

Offset: 0

Alois P. Heinz, Apr 30 2012

The complete bipartite graph K_(n,n) has 2n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2n+1 = A005408(n) coefficients.

			3 example graphs:                     +-----------+
.                 o        o   o      o   o   o   |
.                 |        |\ /|      |\ /|\ /|\ /
.                 |        | X |      | X | X | X
.                 |        |/ \|      |/ \|/ \|/ \
.                 o        o   o      o   o   o   |
.                                     +-----------+
Graph:         K_(1,1)    K_(2,2)      K_(3,3)
Vertices:         2          4            6
Edges:            1          4            9
The complete bipartite graph K_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
  1;
  1,  -1,   0;
  1,  -4,   6,    -3,     0;
  1,  -9,  36,   -75,    78,     -31,       0;
  1, -16, 120,  -524,  1400,   -2236,    1930,     -675, ...
  1, -25, 300, -2200, 10650,  -34730,   75170,  -102545, ...
  1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...
  ...

Alois P. Heinz, Rows n = 0..90, flattened
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Chromatic Polynomial

Columns k=0-2 give: A000012, (-1)*A000290, A083374.
Row sums and last elements of rows give: A000007.
Row lengths give: A005408.
Sums of absolute values of row elements give: A048163(n+1).
T(n,2n-1) = (-1)*A092552(n).
Cf. A008277, A212085, A182368, A185442, A193233, A193277, A193283, A266695, A266972.

P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):
T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):
seq(T(n), n=1..8);

T(n,k) = [q^(2n-k)] Sum_{j=0..n} (q-j)^n * S2(n,j) * Product_{i=0..j-1} (q-i).

T(0,0)=1 prepended by Alois P. Heinz, May 03 2024

A188634 E.g.f.: Sum_{n>=0} (1 - exp(-(n+1)*x))^n/(n+1).

Original entry on oeis.org

1, 1, 4, 46, 1066, 41506, 2441314, 202229266, 22447207906, 3216941445106, 578333776748674, 127464417117501586, 33800841048945424546, 10617398393395844992306, 3898852051843774954576834, 1654948033478889053351543506, 804119629083230641164978005986

Offset: 0

Paul D. Hanna, Dec 28 2012

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 46*x^3/3! + 1066*x^4/4! + 41506*x^5/5! +...
where
A(x) = 1 + (1-exp(-2*x))/2 + (1-exp(-3*x))^2/3 + (1-exp(-4*x))^3/4 + (1-exp(-5*x))^4/5 + (1-exp(-6*x))^5/6 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A099594, A092552, A191908, A266695.

Table[Sum[(-1)^(k+n)*(k+1)^(n-1)*k!*StirlingS2[n, k],{k,0,n}],{n,0,20}]
Table[n!*SeriesCoefficient[Sum[(1-E^(-x*(k+1)))^k/(k+1),{k,0,n}],{x,0,n}],{n,0,20}]  (* Vaclav Kotesovec, Dec 30 2012 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-(k+1)*x+x*O(x^n)))^k/(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=sum(j=0,n, (j+1)^(n-1)*sum(i=0,j, (-1)^(n+j-i)*binomial(j,i)*(j-i)^n))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{j=0..n} (j+1)^(n-1) * Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n.
Ignoring the initial term, equals a diagonal of array A099594, which forms the poly-Bernoulli numbers B(-k,n).
Limit n->infinity a(n)^(1/n)/n^2 = 0.281682... - Vaclav Kotesovec, Dec 30 2012
a(n) = A266695(2*n-1) for n >= 1. - Alois P. Heinz, Apr 17 2024

A212085 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the complete bipartite graph K_(k,k).

Original entry on oeis.org

0, 0, 2, 0, 2, 6, 0, 2, 18, 12, 0, 2, 42, 84, 20, 0, 2, 90, 420, 260, 30, 0, 2, 186, 1812, 2420, 630, 42, 0, 2, 378, 7332, 18500, 9750, 1302, 56, 0, 2, 762, 28884, 127220, 121590, 30702, 2408, 72, 0, 2, 1530, 112740, 825860, 1324470, 583422, 81032, 4104, 90

Offset: 1

Alois P. Heinz, Apr 30 2012

The complete bipartite graph K_(n,n) has 2*n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2*n+1 coefficients.

A(n,k) is the number of pairs of strings of length k over an alphabet of size n such that the strings do not share any letter. - Lin Zhangruiyu, Aug 19 2022

			A(3,1) = 6 because there are 6 3-colorings of the complete bipartite graph K_(1,1): 1-2, 1-3, 2-1, 2-3, 3-1, 3-2.
Square array A(n,k) begins:
   0,   0,    0,      0,       0,        0,         0, ...
   2,   2,    2,      2,       2,        2,         2, ...
   6,  18,   42,     90,     186,      378,       762, ...
  12,  84,  420,   1812,    7332,    28884,    112740, ...
  20, 260, 2420,  18500,  127220,   825860,   5191220, ...
  30, 630, 9750, 121590, 1324470, 13284630, 126657750, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Chromatic Polynomial

Rows n=1-3 give: A000004, A007395, A068293(k+1).
Columns k=1-2 give: A002378(n-1), A091940.
Cf. A008277, A212084, A266695, A282245.

A:= (n, k)-> add(Stirling2(k, j) *mul(n-i, i=0..j-1) *(n-j)^k, j=1..k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

a[n_, k_] := Sum[(-1)^j*(n-j)^k*Pochhammer[-n, j]*StirlingS2[k, j], {j, 1, k}]; Table[a[n-k, k], {n, 1, 11}, {k, n-1, 1, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = Sum_{j=1..k} (n-j)^k * S2(k,j) * Product_{i=0..j-1} (n-i).

A(n,n)/n = A282245(n).

A266858 Number of acyclic orientations of the Turán graph T(n,3).

Original entry on oeis.org

1, 1, 2, 6, 18, 78, 426, 2286, 15402, 122190, 951546, 8724078, 90768378, 928340190, 10779805722, 138779942046, 1759271695338, 24739709631678, 379578822373866, 5743346972756526, 94864142045862282, 1689637343582548590, 29717468115957434586, 563879701735681033998

Offset: 0

Alois P. Heinz, Jan 04 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..465
P. J. Cameron, C. A. Glass, R. U. Schumacher, Acyclic orientations and poly-Bernoulli numbers, arXiv:1412.3685 [math.CO], 2014-2018.
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
Wikipedia, Turán graph

Column k=3 of A267383.
Cf. A182553, A212220, A266695.
Trisection gives A370961.

A[n_, k_] := A[n, k] = Module[{b, l, q}, q = -1; l = Join[Array[Floor[ n/k]&, k - Mod[n, k]], Array[Ceiling[n/k]&, Mod[n, k]]]; b[nn_, j_] := b[nn, j] = If[j==1, (q-nn)^l[[1]] Product[q-i, {i, 0, nn-1}], Sum[b[nn + m, j-1] StirlingS2[l[[j]], m], {m, 0, l[[j]]}]]; Abs[b[0, k]]];
a[n_] := A[n, 3];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Aug 20 2018, after Alois P. Heinz *)

a(n) ~ n! / (2*(1 - log(3/2)) * 3^n * (log(3/2))^(n+1)). - Vaclav Kotesovec, Feb 18 2017

A266972 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n gives the coefficients of the chromatic polynomial of the (n,2)-Turán graph, highest powers first.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -4, 6, -3, 0, 1, -6, 15, -17, 7, 0, 1, -9, 36, -75, 78, -31, 0, 1, -12, 66, -202, 351, -319, 115, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -20, 190, -1080, 3925, -9164, 13186, -10489, 3451, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0

Offset: 0

Alois P. Heinz, Jan 07 2016

The (n,2)-Turán graph is also the complete bipartite graph K_{floor(n/2),ceiling(n/2)}.

			Triangle T(n,k) begins:
  1;
  1,   0;
  1,  -1,   0;
  1,  -2,   1,    0;
  1,  -4,   6,   -3,    0;
  1,  -6,  15,  -17,    7,     0;
  1,  -9,  36,  -75,   78,   -31,    0;
  1, -12,  66, -202,  351,  -319,  115,    0;
  1, -16, 120, -524, 1400, -2236, 1930, -675,  0;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Chromatic Polynomial
Wikipedia, Turán graph

Columns k=0-1 give: A000012, (-1)*A002620.
Main diagonal gives A000007.
Cf. A212084, A266695.

P:= n-> (h-> expand(add(Stirling2(h, j)*mul(q-i,
    i=0..j-1)*(q-j)^(n-h), j=0..h)))(iquo(n, 2)):
T:= n-> (p-> seq(coeff(p, q, n-i), i=0..n))(P(n)):
seq(T(n), n=0..12);

T(n,k) = [q^(n-k)] Sum_{j=0..floor(n/2)} (q-j)^(n-floor(n/2)) * Stirling2(floor(n/2),j) * Product_{i=0..j-1} (q-i).

Sum_{k=0..n} abs(T(n,k)) = A266695(n).

cp's OEIS Frontend

A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).

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A267383 Number A(n,k) of acyclic orientations of the Turán graph T(n,k); square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

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A212084 Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

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A188634 E.g.f.: Sum_{n>=0} (1 - exp(-(n+1)*x))^n/(n+1).

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A212085 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the complete bipartite graph K_(k,k).

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A266858 Number of acyclic orientations of the Turán graph T(n,3).

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